The statistical physics of cities

Abstract

Challenges due to the rapid urbanization of the world — especially in emerging countries — range from an increasing dependence on energy to air pollution, socio-spatial inequalities and environmental and sustainability issues. Modelling the structure and evolution of cities is therefore critical because policy makers need robust theories and new paradigms for mitigating these problems. Fortunately, the increased data available about urban systems opens the possibility of constructing a quantitative ‘science of cities’, with the aim of identifying and modelling essential phenomena. Statistical physics plays a major role in this effort by bringing tools and concepts able to bridge theory and empirical results. This Perspective illustrates this point by focusing on fundamental objects in cities: the distribution of the urban population; segregation phenomena and spin-like models; the polycentric transition of the activity organization; energy considerations about mobility and models inspired by gravity and radiation concepts; CO2 emitted by transport; and finally, scaling that describes how various socio-economical and infrastructures evolve when cities grow.

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Fig. 1: Equilibrium configurations at zero temperature for cities with collective or selfish dynamics.
Fig. 2: Effective speed in human mobility as a function of travel distance.
Fig. 3: Scaling of quantities with city population may depend on the definition of a city.
Fig. 4: Carbon footprint (in millions of tonnes of CO2) versus population for various urban areas.

References

  1. 1.

    United Nations. World Urbanization Prospects. https://esa.un.org/unpd/wup/ (2018).

  2. 2.

    Chorley, R. & Haggett, P. Models in geography (1967).

  3. 3.

    Fujita, M., Krugman, P. & Venables, A. The Spatial Economy: Cities, Regions, and International Trade (MIT Press, Cambridge, 2001).

    Google Scholar 

  4. 4.

    Batty, M. The New Science of Cities (MIT Press, Cambridge, 2013).

    Google Scholar 

  5. 5.

    Barthelemy, M. The Structure and Dynamics of Cities (Cambridge University Press, Cambridge, 2016).

    Google Scholar 

  6. 6.

    Kivelson, S. & Kivelson, S. Understanding complexity. Nat. Phys. 14, 426 (2018).

  7. 7.

    Newman, P. & Kenworthy, J. Gasoline consumption and cities: a comparison of US cities with a global survey. J. Am. Plan. Assoc. 55, 24–37 (1989).

    Google Scholar 

  8. 8.

    Vespignani, A. Predicting the behavior of techno-social systems. Science 325, 425–428 (2009).

    ADS  MathSciNet  MATH  Google Scholar 

  9. 9.

    Castellano, C., Fortunato, S. & Loreto, V. Statistical physics of social dynamics. Rev. Mod. Phys. 81, 591 (2009).

    ADS  Google Scholar 

  10. 10.

    Dadashpoor, H. & Yousefi, Z. Centralization or decentralization? A review on the effects of information and communication technology on urban spatial structure. Cities 78, 194–205 (2018).

    Google Scholar 

  11. 11.

    Blondel, V., Decuyper, A. & Krings, G. A survey of results on mobile phone datasets analysis. EPJ Data Sci. 4, 10 (2015).

    Google Scholar 

  12. 12.

    Barbosa, H., Barthelemy, M., Ghoshal, G., James, C., Lenormand, M., Louail, T., Menezes, R., Ramasco, J., Simini, F. & Tomasini, M. Human mobility: models and applications. Phys. Rep. 734, 1–74 (2018).

  13. 13.

    Zipf, G. Human Behavior and the Principle of Least Effort: An Introduction to Human Ecology (Addison-Wesley, Cambridge, MA, 1949).

    Google Scholar 

  14. 14.

    Batty, M. Rank clocks. Nature 444, 592 (2006).

    ADS  Google Scholar 

  15. 15.

    Malevergne, Y., Pisarenko, V. & Sornette, D. Testing the Pareto against the lognormal distributions with the uniformly most powerful unbiased test applied to the distribution of cities. Phys. Rev. E 83, 036111 (2011).

    ADS  Google Scholar 

  16. 16.

    Soo, K. Zipf’s Law for cities: a cross-country investigation. Reg. Sci. Urban Econ. 35, 239–263 (2005).

    Google Scholar 

  17. 17.

    Cristelli, M. & Batty, M. P. L. There is more than a power law in Zipf. Sci. Rep. 2, 812 (2012).

    Google Scholar 

  18. 18.

    Gibrat, R. Les inégalités économiques: applications: aux inégalités des richesses, à la concentration des entreprises, aux populations des villes, aux statistiques des familles, etc: d’une loi nouvelle: la loi de l’effet proportionnel (Librairie du Recueil Sirey, 1931).

  19. 19.

    Marsili, M. & Zhang, Y. Interacting individuals leading to Zipf’s law. Phys. Rev. Lett. 80, 2741 (1998).

    ADS  Google Scholar 

  20. 20.

    Gabaix, X. Zipf’s law for cities: an explanation. Quart. J. Econ. 114, 739–767 (1999).

    MATH  Google Scholar 

  21. 21.

    Levy, M. & Solomon, S. Power laws are logarithmic Boltzmann laws. Int. J. Mod. Phys. C 7, 595–601 (1996).

    ADS  Google Scholar 

  22. 22.

    Sornette, D. & Cont, R. Convergent multiplicative processes repelled from zero: power laws and truncated power laws. J. Phys. I 7, 431–444 (1997).

    Google Scholar 

  23. 23.

    Bouchaud, J.-P. & Mézard, M. Wealth condensation in a simple model of economy. Physica A 282, 536–545 (2000).

    ADS  Google Scholar 

  24. 24.

    De Nadai, M., Staiano, J., Larcher, R., Sebe, N., Quercia, D. & Lepri, B. The death and life of great Italian cities: a mobile phone data perspective. in 25th International World Wide Web Conferences (2016).

  25. 25.

    Sulis, P., Manley, E., Zhong, C. & Batty, M. Using mobility data as proxy for measuring urban vitality. J. Spat. Inf. Sci. 16, 137–162 (2018).

    Google Scholar 

  26. 26.

    Jacobs, J. The Death and Life of Great American Cities (Vintage, New York, NY, 1961).

    Google Scholar 

  27. 27.

    D’Silva, K., Noulas, A., Musolesi, M., Mascolo, C. & Sklar, M. If I build it, will they come?: Predicting new venue visitation patterns through mobility data. in Proceedings of the 25th ACM SIGSPATIAL International Conference on Advances in Geographic Information Systems (2017).

  28. 28.

    Bouchaud, J.-P. Crises and collective socio-economic phenomena: simple models and challenges. J. Stat. Phys. 151, 567–606 (2013).

    ADS  MathSciNet  MATH  Google Scholar 

  29. 29.

    Venerandi, A., Zanella, M., Romice, O., Dibble, J. & Porta, S. Form and urban change–An urban morphometric study of five gentrified neighbourhoods in London. Environment and Planning B: Urban Analytics and City. Science 44, 1056–1076 (2017).

    Google Scholar 

  30. 30.

    Schelling, T. Dynamic models of segregation. J. Math. Sociol. 1, 143–186 (1971).

    MATH  Google Scholar 

  31. 31.

    Grauwin, S., Bertin, E., Lemoy, R. & Jenson, P. Competition between collective and individual dynamics. Proc. Natl Acad. Sci. USA 106, 20622–20626 (2009).

  32. 32.

    Gauvin, L., Vannimenus, J. & Nadal, J.-P. Phase diagram of a Schelling segregation model. Eur. Phys. J. B 70, 293–304 (2009).

    ADS  Google Scholar 

  33. 33.

    Dall’Asta, L., Castellano, C. & Marsili, M. Statistical physics of the Schelling model of segregation. J. Stat. Mech.: Theory Exp. 7, L07002 (2008).

    Google Scholar 

  34. 34.

    Jensen, P., Matreux, T., Cambe, J., Larralde, H. & Bertin, E. Giant catalytic effect of altruists in Schelling’s segregation model. Phys. Rev. Lett. 120, 208301 (2018).

    ADS  Google Scholar 

  35. 35.

    Vinković, D. & Kirman, A. A physical analogue of the Schelling model. Proc. Natl. Acad. Sci. USA. 103, 19261–19265 (2006).

    ADS  Google Scholar 

  36. 36.

    Bertaud, A. & Malpezzi, S. The spatial distribution of population in 48 world cities: implications for economies in transition. in World Bank Report (2003).

  37. 37.

    Anas, A., Arnott, R. & Small, K. Urban spatial structure. J. Econ. Lit. 36, 1426–1464 (1998).

    Google Scholar 

  38. 38.

    Ratti, C., Frenchman, D., Pulselli, R. & Williams, S. Mobile landscapes: using location data from cell phones for urban analysis. Environ. Plan. B 33, 727–748 (2006).

    Google Scholar 

  39. 39.

    Louail, T. et al. From mobile phone data to the spatial structure of cities. Sci. Rep. 4, 5276 (2014).

    Google Scholar 

  40. 40.

    Louail, T. et al. Uncovering the spatial structure of mobility networks. Nat. Commun. 6, 6007 (2015).

    Google Scholar 

  41. 41.

    Louf, R. & Barthelemy, M. Modeling the polycentric transition of cities. Phys. Rev. Lett. 111, 198702 (2013).

    ADS  Google Scholar 

  42. 42.

    Samaniego, H. & Moses, M. Cities as organisms: allometric scaling of urban road networks. J. Transp. Land Use 1, 21–39 (2008).

    Google Scholar 

  43. 43.

    Zhong, C. et al. Revealing centrality in the spatial structure of cities from human activity patterns. Urban Stud. 54, 437–455 (2017).

    Google Scholar 

  44. 44.

    Zhang, X., Xu, Y., Tu, W. & Ratti, C. Do different datasets tell the same story about urban mobility — A comparative study of public transit and taxi usage. J. Transp. Geogr. 70, 78–90 (2018).

    Google Scholar 

  45. 45.

    Krugman, P. The Self-Organizing Economy (CIMMYT, 1996).

  46. 46.

    Fujita, M. & Ogawa, H. Multiple equilibria and structural transition of non-monocentric urban configurations. Reg. Sci. Urban Econ. 12, 161–196 (1982).

    Google Scholar 

  47. 47.

    Louf, R. & Barthelemy, M. How congestion shapes cities: from mobility patterns to scaling. Sci. Rep. 4, 5561 (2014).

    ADS  Google Scholar 

  48. 48.

    Wigner, E. Characteristic vectors of bordered matrices with infinite dimensions. Ann. Math. 62, 548–564 (1955).

    MathSciNet  MATH  Google Scholar 

  49. 49.

    Branston, D. Link capacity functions: a review. Transp. Res. 10, 223–236 (1976).

    Google Scholar 

  50. 50.

    Huntsinger, L. & Rouphail, N. Bottleneck and queuing analysis: calibrating volume–delay functions of travel demand models. Transp. Res. Rec. 2255, 117–124 (2011).

    Google Scholar 

  51. 51.

    Varga, L., Kovács, A., Tóth, G., Papp, I. & Néda, Z. Further we travel the faster we go. PloS One 11, e0148913 (2016).

    Google Scholar 

  52. 52.

    Gallotti, R., Bazzani, A., Rambaldi, S. & Barthelemy, M. A stochastic model of randomly accelerated walkers for human mobility. Nat. Commun. 7, 12600 (2016).

    ADS  Google Scholar 

  53. 53.

    Zahavi, Y. Traveltime budgets and mobility in urban areas (1974).

  54. 54.

    Marchetti, C. Anthropological invariants in travel behavior. Technol. Forecast. Soc. change 47, 75–88 (1994).

    Google Scholar 

  55. 55.

    Mokhtarian, P. & Chen, C. TTB or not TTB, that is the question: a review and analysis of the empirical literature on travel time (and money) budgets. Transp. Res. Part A 38, 643–675 (2004).

    Google Scholar 

  56. 56.

    Levinson, D. & Wu, Y. The rational locator reexamined: Are travel times still stable?. Transportation 32, 187–202 (2005).

    Google Scholar 

  57. 57.

    Zhu, S. & Levinson, D. Do people use the shortest path? An empirical test of Wardrop’s first principle. PLoS One 10, e0134322 (2015).

    Google Scholar 

  58. 58.

    Tang, W. & Levinson, D. Deviation between actual and shortest travel time paths for commuters. J. Transp. Eng. Part A 144, 04018042 (2018).

    Google Scholar 

  59. 59.

    Lima, A., Stanojevic, R., Papagiannaki, D., Rodriguez, P. & González, M. Understanding individual routing behaviour. J. R. Soc. Interface 13, 20160021 (2016).

    Google Scholar 

  60. 60.

    Noulas, A., Scellato, S., Lambiotte, R., Pontil, M. & Mascolo, C. A tale of many cities: universal patterns in human urban mobility. PloS One 7, e37027 (2012).

    ADS  Google Scholar 

  61. 61.

    Çolak, S., Lima, A. & González, M. Understanding congested travel in urban areas. Nat. Commun. 7, 10793 (2016).

    ADS  Google Scholar 

  62. 62.

    Solé-Ribalta, A., Gómez, S. & Arenas, A. Decongestion of urban areas with hotspot pricing. Netw. Spat. Econ. 18, 33–50 (2018).

    MathSciNet  Google Scholar 

  63. 63.

    Kolbl, R. & Helbing, D. Energy laws in human travel behaviour. New J. Phys. 5, 48 (2003).

    ADS  Google Scholar 

  64. 64.

    Zipf, G. The P 1 P 2/D hypothesis: on the intercity movement of persons. Am. Sociol. Rev. 11, 677–686 (1946).

    Google Scholar 

  65. 65.

    Erlander, S. & Stewart, N. The Gravity Model in Transportation Analysis: Theory and Extensions (Vsp, 1990).

  66. 66.

    Lenormand, M., Huet, S., Gargiulo, F. & Deffuant, G. A universal model of commuting networks. PloS One 7, e45985 (2012).

    ADS  Google Scholar 

  67. 67.

    Simini, F., González, M., Maritan, A. & Barabási, A. A universal model for mobility and migration patterns. Nature 484, 96 (2012).

    ADS  Google Scholar 

  68. 68.

    Wilson, A. Complex Spatial Systems: The Modelling Foundations of Urban and Regional Analysis (Routledge, 2014).

  69. 69.

    Varga, L., Tóth, G. & Néda, Z. Commuting patterns: the flow and jump model and supporting data. EPJ Data Sci. 7, 37 (2018).

    Google Scholar 

  70. 70.

    Lenormand, M., Bassolas, A. & Ramasco, J. Systematic comparison of trip distribution laws and models. J. Transp. Geogr. 51, 158–169 (2016).

    Google Scholar 

  71. 71.

    Carra, G., Mulalic, I., Fosgerau, M. & Barthelemy, M. Modelling the relation between income and commuting distance. J. R. Soc. Interface 13, 20160306 (2016).

    Google Scholar 

  72. 72.

    Yang, X., Belyi, A., Bojic, I. & Ratti, C. Human mobility and socioeconomic status: analysis of Singapore and Boston. Comput. Environ. Urban Syst. 72, 51–67 (2018).

  73. 73.

    Pumain, P. Scaling Laws and Urban Systems (Santa Fe Institute, 2004).

  74. 74.

    Bettencourt, L., Lobo, J. & Youn, H. The hypothesis of urban scaling: formalization, implications and challenges. https://arxiv.org/abs/1301.5919 (2013).

  75. 75.

    Bettencourt, L., Lobo, J., Helbing, D., Kühnert, C. & West, G. Growth, innovation, scaling, and the pace of life in cities. Proc. Natl Acad. Sci. USA 104, 7301–7306 (2007).

    ADS  Google Scholar 

  76. 76.

    Schläpfer, M. et al. The scaling of human interactions with city size. J. R. Soc. Interface 11, 20130789 (2014).

    Google Scholar 

  77. 77.

    Bettencourt, L. The origins of scaling in cities. Science 340, 1438–1441 (2013).

    ADS  MathSciNet  MATH  Google Scholar 

  78. 78.

    Arcaute, E. et al. Constructing cities, deconstructing scaling laws. J. R. Soc. Interface 12, 20140745 (2015).

    Google Scholar 

  79. 79.

    Leitao, J., Miotto, J., Gerlach, M. & Altmann, E. Is this scaling nonlinear?. R. Soc. Open Sci. 3, 150649 (2016).

    ADS  MathSciNet  Google Scholar 

  80. 80.

    Dyson, F. The key to everything. The New York Review of Books, 10 May (2018).

  81. 81.

    Rozenfeld, H. et al. Laws of population growth. Proc. Natl Acad. Sci. USA 105, 18702–18707 (2008).

    ADS  Google Scholar 

  82. 82.

    Depersin, J. & Barthelemy, M. From global scaling to the dynamics of individual cities. Proc. Natl. Acad. Sci. USA 115, 2317–2322 (2018).

    ADS  Google Scholar 

  83. 83.

    Bouchaud, J.-P., Cugliandolo, L., Kurchan, J. & Mezard, M. in Spin Glasses and Random Fields (ed. Young, A.) (Singapore, World Scientific, 1998).

  84. 84.

    Thisse, J.-F. The New Science Of Cities by Michael Batty: the opinion of an economist. J. Econ. Lit. 52, 805–819 (2014).

    Google Scholar 

  85. 85.

    Meirelles, J., Neto, C. R., Ferreira, F. F., Ribeiro, F. L. & Binder, C. Evolution of urban scaling: evidence from Brazil. Preprint at http://arxiv.org/abs/1807.02292 (2018).

  86. 86.

    Canadell, J., C. P., Le QuéRé, C., Dhakal, S. & Raupach, M. The human perturbation of the carbon cycle (UNESCO-SCOPE-UNEP, Paris, 2009).

  87. 87.

    Moran, D. et al. Carbon footprints of 13 000 cities. Environ. Res. Lett. 13, 064041 (2018).

    ADS  Google Scholar 

  88. 88.

    Velasco, E. & Roth, M. Cities as net sources of CO2: Review of atmospheric CO2 exchange in urban environments measured by eddy covariance technique. Geogr. Compass 4, 1238–1259 (2010).

    Google Scholar 

  89. 89.

    Glaeser, E. & Kahn, M. The greenness of cities: carbon dioxide emissions and urban development. J. Urban Econ. 67, 404–418 (2010).

    Google Scholar 

  90. 90.

    Louf, R. & Barthelemy, M. Scaling: lost in the smog. Environ. Plan. B 41, 767–769 (2014).

    Google Scholar 

  91. 91.

    Fragkias, M., Lobo, J., Strumsky, D. & Seto, K. Does size matter? Scaling of CO2 emissions and US urban areas. PloS One 8, e64727 (2013).

    ADS  Google Scholar 

  92. 92.

    Oliveira, E., Andrade, J. & Makse, H. Large cities are less green. Sci. Rep. 4, 4235 (2014).

    ADS  Google Scholar 

  93. 93.

    Rybski, D. et al. Cities as nuclei of sustainability? Environment and Planning B: Urban Analytics and City. Science 44, 425–440 (2017).

    Google Scholar 

  94. 94.

    Carantino, B. & Lafourcade, M. The carbon “carprint” of suburbanization: new evidence from French Cities. CEPR Discussion Papers (2018).

  95. 95.

    Verbavatz, V. & Barthelemy, M. Critical factors for mitigating car traffic in cities. Preprint at https://arxiv.org/abs/1901.01386 (2019).

  96. 96.

    Creutzig, F., Baiocchi, G., Bierkandt, R., Pichler, P. & Seto, K. Global typology of urban energy use and potentials for an urbanization mitigation wedge. Proc. Natl. Acad. Sci. USA 112, 6283–6288 (2015).

    ADS  Google Scholar 

  97. 97.

    Levinson, D. & Yerra, B. Self-organization of surface transportation networks. Transp. Sci. 40, 179–188 (2006).

    Google Scholar 

  98. 98.

    Strano, E., Nicosia, V., Latora, V., Porta, S. & Barthelemy, M. Elementary processes governing the evolution of road networks. Sci. Rep. 2, 296 (2012).

    ADS  Google Scholar 

  99. 99.

    Kivelä, M. et al. Multilayer networks. J. Complex Netw. 2, 203–271 (2014).

    Google Scholar 

  100. 100.

    Gallotti, R. & Barthelemy, M. Anatomy and efficiency of urban multimodal mobility. Sci. Rep. 4, 6911 (2014).

    ADS  Google Scholar 

  101. 101.

    Sobstyl, J., Emig, T., Qomi, M., Ulm, F. & Pellenq, R. Role of city texture in urban heat islands at nighttime. Phys. Rev. Lett. 120, 108701 (2018).

    ADS  Google Scholar 

  102. 102.

    Bunde, A. & Havlin, S. (eds). Fractals and Disordered Systems (Springer Science & Business Media, 1996).

  103. 103.

    Batty, M. & Longley, P. Fractal Cities: A Geometry of Form and Function (Academic Press, 1994).

  104. 104.

    Tannier, C. & Pumain, D. Fractals in urban geography: a theoretical outline and an empirical example. Cybergeo (2005).

  105. 105.

    Benguigui, L. & Daoud, M. Is the suburban railway system a fractal?. Geogr. Anal. 23, 362 (1991).

    Google Scholar 

  106. 106.

    Witten, T. & Sander, L. Diffusion-limited aggregation. Phys. Rev. B 27, 5686 (1983).

    ADS  MathSciNet  Google Scholar 

  107. 107.

    Makse, H., Havlin, S. & Stanley, H. Modelling urban growth patterns. Nature 377, 608 (1995).

    ADS  Google Scholar 

  108. 108.

    Makse, H., Andrade, J., Batty, M., Havlin, S. & Stanley, H. Modeling urban growth patterns with correlated percolation. Phys. Rev. E 58, 7054 (1998).

    ADS  Google Scholar 

  109. 109.

    Census Bureau. https://www.census.gov/ (2018).

  110. 110.

    European Commission. https://ec.europa.eu/eurostat/ (2018).

  111. 111.

    Labs, N. Y. P. L. http://www.nypl.org/collections/labs (2015).

  112. 112.

    M. Barthelemy. Quanturb data page. https://www.quanturb.com/data.html (2018).

  113. 113.

    Geohistorical Data Research Group. http://www.geohistoricaldata.org (2018).

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Barthelemy, M. The statistical physics of cities. Nat Rev Phys 1, 406–415 (2019). https://doi.org/10.1038/s42254-019-0054-2

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